Symmetry transformations for square sliced three-way arrays, with applications to their typical rank
type: Item
instance of: scholarly article
title: Symmetry transformations for square sliced three-way arrays, with applications to their typical rank
zbMATH Open document ID: 1104.15022
author: Jos M. F. ten Berge
author: Alwin Stegeman
published in: Linear Algebra and its Applications
publication date: 2006-11-15T00:00:00Z
review text: The typical tensorial rank of three-way arrays over algebraically closed fields has been much studied. In this paper, the tensorial rank over the real field is examined. The possible values of typical rank of some \(p \times q \times r\) arrays are given. Especially, the typical rank of arrays with square slices, i.~e., with \(q = r\) are analyzed. When \(p = 2\), square slice arrays have typical rank \([q, q+1]\). Special attention is given to square cases with \(p > 2\). The bounds for the typical rank of such arrays are derived and it is proved that the typical rank of the ``cubic'' \(3 \times 3 \times 3\) array is \(5\). The main result also allows, for \(p \times q \times r\) arrays with \(p > 2\), to discard \(q + 1\) as a value of positive probability, except when \(p = 3\) and \(q = 2\). This is done by introducing a symmetry criterion.
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 5072863
zbMATH Keywords: three-way rank
zbMATH Keywords: typical tensorial rank
zbMATH Keywords: Indscal
zbMATH Keywords: Candecomp
zbMATH Keywords: Parafac
zbMATH Keywords: three-mode component analysis
MaRDI profile type: Publication
full work available at URL: j.laa.2006.02.002
OpenAlex ID: W2103665081
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DOI: 10.1016/J.LAA.2006.02.002
description: scientific article; zbMATH DE number 5072863